- Research
- Open access
- Published:
On the reciprocal products of generalized Fibonacci sequences
Journal of Inequalities and Applications volume 2022, Article number: 154 (2022)
Abstract
In this paper, we use the properties of error estimation and the analytic method to study the reciprocal products of the bi-periodic Fibonacci sequence, the bi-periodic Lucas sequence, and the mth-order linear recursive sequence.
1 Introduction
The so-called Fibonacci sequence \(\{ F_{n} \} \) and Lucas sequence \(\{ L_{n} \} \) are defined by
and
The Fibonacci and Lucas sequences have many interesting properties and applications [1]. In addition, in [2], Ohtsuka and Nakamura considered the partial infinite sums of reciprocal Fibonacci sequence and proved that:
and
where \(\lfloor \cdot \rfloor \) (the floor function) denotes the greatest integer less than or equal to x.
Many authors have studied the Fibonacci and Lucas sequences by changing initial conditions or recursive relations. For instance, for any two nonzero real numbers a and b, Edson and Yayenie [3] introduced the bi-periodic Fibonacci sequence \(\{ f_{n} \} \) as:
For \(a=b=1\), \(\{ f_{n} \} \) reduces to the Fibonacci sequence \(\{ F_{n} \} \). If \(a=b=k\), then \(\{ f_{n} \} \) becomes the k-Fibonacci sequence \(\{ q_{n} \} \) defined in [4], etc. Similarly, for any two nonzero real numbers a and b, Bilgici [5] introduced the bi-periodic Lucas sequence \(\{ l_{n} \} \) as:
For \(a=b=1\), \(\{ l_{n} \} \) reduces to the Lucas sequence \(\{ L_{n} \} \). If \(a=b=k\), then \(\{ l_{n} \} \) becomes the k-Lucas sequence \(\{ p_{n} \} \) defined in [6]. In [7], Tan and Leung considered a generalization of Horadam sequence \(\{ w_{n} \}\), which is defined by the recurrence relation
with arbitrary initial conditions \(w_{0} \), \(w_{1} \) and nonzero real numbers a, b, and c. In [8], Tan considered the sequence \(\{ w_{n} \} \) when \(c=1\). In [9], Ramírez and Sirvent introduced a q-bi-periodic Fibonacci sequence by
with initial conditions \(F_{0} ^{(a,b)} (q,s)=0\) and \(F_{1} ^{(a,b)} (q,s) =1\) and nonzero real numbers a, b, q and s. Motivated by [9], in [10] Tan introduced a q-bi-periodic Lucas sequence by
with initial conditions \(L_{0} ^{(a,b)} (q,s)=2\) and \(L_{1} ^{(a,b)} (q,s) =q\), and nonzero real numbers a, b,q and s.
In [11], Holliday and Komatsu obtained the infinite sums of the reciprocal of k-Fibonacci sequence \(\{ q_{n} \}\). In [12], Basbük and Yazlik obtained the infinite sums of the reciprocal of the bi-periodic Fibonacci sequence \(\{ f_{n} \} \). Various authors studied the infinite sums of the reciprocal of the other famous sequences [13–15].
Recently, some authors studied the nearest integer of the sums of reciprocal of some linear recurrence sequences. In [16], Komatsu proved that there exists a positive integer \(n_{1} \) such that:
where \(\{ q_{n} \} \) is the k-Fibonacci sequence. \(\Vert \cdot \Vert \) denotes the nearest integer. Specifically, suppose that \(\Vert x \Vert = \lfloor x+\frac{1}{2} \rfloor \).
On the other hand, Wu and Zhang [17] considered an mth-order linear recursive sequence \(\{ u_{n} \} \) defined by
where initial values \(u_{i}\in N\) for \(0\le i < m\), at least one of them is different from zero, and \(x_{1}, x_{2}, \dots , x_{m} \) are positive integers. The characteristic polynomial of the sequence \(\{ u_{n} \} \) is given by
For \(m=2\), \(x_{1}=x_{2}=1\) and initial values \(u_{0}=0\), \(u_{1}=1\), \(\{ u_{n} \} \) reduces to the Fibonacci sequence. If \(m=2\), \(x_{1}=x_{2}=1\) and initial values \(u_{0}=2\), \(u_{1}=1\), then \(\{ u_{n} \} \) becomes the Lucas sequence.
In addition, they proved that there exists a positive integer \(n_{2} \) such that:
for any positive integers \(x_{1}\ge x_{2}\ge \cdots \ge x_{m}\ge 1\). For more the nearest integer of the sums of reciprocal of the recurrence sequence studies, see [18–21]. Specifically, in [19], Trojorský considered finding a sequence that is “asymptotically equivalent” to partial infinite sums and proved that
are asymptotically equivalent, where \(P ( z )\in C [ z ]\) is a non-constant polynomial. Specifically, we say that two sequences \(\{ G_{n} \}\) and \(\{ H_{n} \}\) are called “asymptotically equivalent” if \(\{ G_{n} \} / \{ H_{n} \}\) tends to 1 as \(n \rightarrow \infty \).
In addition to the study of the infinite reciprocal sums of recursive sequence, we can also consider the infinite reciprocal products of recursive sequence. In 2006, Wu [22] studied the partial infinite products of \(\frac{q_{k}^{i} -1 }{q_{k}^{i} } \). He used the element method and the properties of the floor function and proved that
and
where \(\{ q_{n} \} \) is the k-Fibonacci sequence. For more the partial infinite products of the other sequences, see [23, 24].
Inspired by [19], in this paper, we apply a different research method from the previous one and use the properties of error estimation and the analytic method to study the reciprocal products of \(\{ f_{n} \} \), \(\{ l_{n} \} \) and \(\{ u_{n} \} \). We derive some sequences that are asymptotically equivalent to reciprocal products including \(\{ f_{n} \} \), \(\{ l_{n} \} \) and \(\{ u_{n} \} \). Our main results are the following:
Theorem 1
Let \(\{ f_{n} \} \) be the bi-periodic Fibonacci sequence, and \(\{ l_{n} \} \) be the bi-periodic Lucas sequence. For positive integers a and b with \(a\ge 1\), \(b\ge 1\), the sequences
are asymptotically equivalent, and the sequences
are asymptotically equivalent.
Corollary 1
We obtain the infinite products of the reciprocal of the k-Fibonacci sequence \(q_{n}\) and k-Lucas sequence \(p_{n}\), when \(a=b=k\). Then, the sequences
are asymptotically equivalent, and the sequences
are asymptotically equivalent.
Theorem 2
Let \(\{ u_{n} \} \) be an mth-order linear recursive sequence with any positive integers \(x_{1}\ge x_{2}\ge \cdots \ge x_{m}\ge 1\). Then, the sequences
are asymptotically equivalent.
2 Proof of the theorems
To complete the proof of our theorems, we need the following:
Lemma 1
([3, 5], Generalized Binet’s formula)
The terms of the bi-periodic Fibonacci sequence \(\{ f_{n} \} \), and bi-periodic Lucas sequence \(\{ l_{n} \} \) are given by
and
where \(\alpha =\frac{ab+\sqrt{a^{2}b^{2}+4ab } }{2} \) and \(\beta =\frac{ab-\sqrt{a^{2}b^{2}+4ab } }{2} \), i.e. α and β are roots of the equation \(x^{2} -abx-ab=0\). It is obvious that \(\alpha > 1\) and \(-1<\beta < 0\) with \(a\ge 1\), \(b\ge 1\). In addition, \(\zeta ( n ) \) is the parity function, such that \(\zeta ( n ) =0\) if n is even and \(\zeta ( n ) =1\) if n is odd.
Lemma 2
Let \(\{ f_{n} \} \) be the bi-periodic Fibonacci sequence defined by (1), and \(\{ l_{n} \} \) be the bi-periodic Lucas sequence defined by (2). Then, we have
where \(c=\frac{a}{\alpha -\beta }\), \(d=\frac{1}{\alpha -\beta }\), and
Proof
By Lemma 1, we can easily prove it. □
Lemma 3
([17])
Let \(\{u_{n} \}\) be an mth-order linear recursive sequence defined by (3). The coefficients of the characteristic polynomial \(\psi ( y )\) are satisfied that \(x_{1}\ge x_{2}\ge \cdots \ge x_{m}\ge 1\). Then, the closed formula of \(\{u_{n} \}\) is given by
where \(s>0\), \(t>1\), γ is the positive real zero of \(\psi ( y )\) for \(x_{1}<\gamma <x_{1}+1 \), and “\(\mathcal{O}\)” (the Landau symbol) denotes if \(g ( x ) > 0\) for all \(x\ge a\), we write \(f ( x ) =\mathcal{O} ( g ( x ) )\) to mean that the quotient \(f ( x ) /g ( x ) \) is bounded for \(x\ge a\).
Lemma 4
Let a, b, c, d, α, and β be defined by Lemma 1or Lemma 2and s, γ, and t be defined by Lemma 3. Then, we have
Proof
We shall prove only (6) in Lemma 4, and other identities are proved similarly. The identity \(ab=-\alpha \beta \) now yield \(\vert \beta \vert < ( ab )^{\frac{1}{2} }= ( - \alpha \beta ) ^{\frac{1}{2} }< \alpha \), where \(\alpha > 1\) and \(-1<\beta < 0\). First, we prove the following equation
We prove (9) by mathematical induction. When \(m=1\),
When \(m=2\),
That is, (9) is true for \(m=1\) or \(m=2\). Suppose that for any integer m, we have
Then, for \(m+1\), we have
Taking \(m \rightarrow \infty \), we have
which completes the proof. □
Proof of Theorem 1
We shall prove only (4) in Theorem 1, and the identity (5) is proved similarly. From the geometric series as \(\epsilon \rightarrow 0\), we find
If n is even, with \(n\ge 2\). Using Lemma 2, we have
By Lemma 4, we obtain
Taking the reciprocal of this expression yields
where \(\vert \beta \vert < ( ab )^{\frac{1}{2} }\) yields
as \(n \rightarrow \infty \). In addition, as \(( ab )^{\frac{1}{2} }< \alpha \), we obtain
as \(n \rightarrow \infty \).
If n is odd, with \(n\ge 1\). Using Lemma 2, we have
By Lemma 4, we obtain
Taking the reciprocal of this expression yields
where \(\vert \beta \vert < ( ab )^{\frac{1}{2} }\) yields
as \(n \rightarrow \infty \). In addition, as \(( ab )^{\frac{1}{2} }< \alpha \), we obtain
as \(n \rightarrow \infty \), which completes the proof. □
Proof of Theorem 2
Using Lemma 3, we have
By Lemma 4, we obtain
Taking the reciprocal of this expression yields
which yields
as \(n \rightarrow \infty \), which completes the proof. □
3 Discussion
In this paper, we obtain the sequences that are asymptotically equivalent to reciprocal products of \(\frac{f_{k}-1 }{f_{k}}\), \(\frac{l_{k}-1 }{l_{k}}\) and \(\frac{u_{k}-1 }{u_{k}}\), where \(\{ f_{n} \} \) denotes the bi-periodic Fibonacci sequence, \(\{ l_{n} \} \) denotes the bi-periodic Lucas sequence, and \(\{ u_{n} \} \) denotes an mth-order linear recursive sequence. For any positive integers j, an open problem is whether there exists the similar identities for the infinity products of \(\frac{f_{k}^{j} -1 }{f_{k}^{j} }\), \(\frac{l_{k}^{j} -1 }{l_{k}^{j} }\) and \(\frac{u_{k}^{j} -1 }{u_{k}^{j} }\).
Availability of data and materials
All of the material is owned by the authors and no permissions are required.
References
Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, New York (2001)
Ohtsuka, H., Nakamura, S.: On the sum of reciprocal Fibonacci numbers. Fibonacci Q. 46–47, 153–159 (2008)
Edson, M., Yayenie, O.: A new generalization of Fibonacci sequence and extended Binet’s formula. Integers 9, 639–654 (2009)
Falcon, S.: On the Fibonacci k-numbers. Chaos Solitons Fractals 32, 1615–1624 (2007)
Bilgici, G.: Two generalizations of Lucas sequence. Appl. Math. Comput. 245, 526–538 (2014)
Falcon, S.: On the k-Lucas numbers. Int. J. Contemp. Math. Sci. 6, 1039–1050 (2011)
Tan, E., Leung, H.-H.: Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences. Adv. Differ. Equ. 2020, 26 (2020)
Tan, E.: Some properties of bi-periodic Horadam sequences. Notes Number Theory Discrete Math. 23(4), 56–65 (2017)
Ramírez, J.L., Sirvent, V.F.: A q-analoque of the biperiodic Fibonacci sequence. J. Integer Seq. 19(2), 3 (2016)
Tan, E.: A Q-analog of the BI-periodic Lucas sequence. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 67(2), 220–228 (2018)
Holliday, S., Komatsu, T.: On the sum of reciprocal generalized Fibonacci numbers. Integers 11, 441–455 (2011)
Basbük, M., Yazlik, Y.: On the sum of reciprocal of generalized bi-periodic Fibonacci numbers. Miskolc Math. Notes 17, 35–41 (2016)
Zhang, W., Wang, T.: The infinite sum of reciprocal Pell numbers. Appl. Math. Comput. 218, 6164–6167 (2012)
Choi, G., Choo, Y.: On the reciprocal sums of products of Fibonacci and Lucas numbers. Filomat 32, 2911–2920 (2018)
Choi, G., Choo, Y.: On the reciprocal sums of square of generalized bi-periodic Fibonacci numbers. Miskolc Math. Notes 19, 201–209 (2018)
Komatsu, T.: On the nearest integer of the sum of reciprocal Fibonacci numbers. Aport. Mat. Investig. 20, 171–184 (2011)
Wu, Z., Han, Z.: On the reciprocal sums of higher-order sequences. Adv. Differ. Equ. 2013, 189 (2013)
Wu, Z., Zhang, J.: On the higher power sums of reciprocal higher-order sequences. Sci. World J. 2014, 521358 (2014)
Trojovský, P.: On the sum of reciprocal of polynomial applied to higher order recurrences. Mathematics 7(7), 638 (2019)
Zhang, H., Wu, Z.: On the reciprocal sums of the generalized Fibonacci sequences. Adv. Differ. Equ. 2013, 377 (2013)
Kiliç, E., Arikan, T.: More on the infinite sum of reciprocal Fibonacci, Pell and higher order recurrences. Appl. Math. Comput. 219, 7783–7788 (2013)
Wu, Z.: Several identities relating to reciprocal products of generalized Fibonacci numbers. J. Northwest Univ. Nat. Sci. 46(3), 317–320 (2016)
Wu, Z.: On the study of some identities related to Riemann zeta function. J. Shaanxi Normal Univ. Nat. Sci. Ed. 46(2), 26–29 (2018)
Jiang, Y., Wang, T.: Some identities involving the reciprocal products of the Pell numbers. J. Shaanxi Normal Univ. Nat. Sci. Ed. 45(4), 23–27 (2017)
Acknowledgements
The authors express their gratitude to the referee for very helpful and detailed comments.
Funding
Supported by the National Natural Science Foundation of China (Grant No. 11701448).
Author information
Authors and Affiliations
Contributions
Du Tingting wrote the main manuscript text and Wu Zhengang examined the manuscript, and all the authors reviewed the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Du, T., Wu, Z. On the reciprocal products of generalized Fibonacci sequences. J Inequal Appl 2022, 154 (2022). https://doi.org/10.1186/s13660-022-02889-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-022-02889-8